Long answer: This is what I've gathered from my curious research on that stupid little thing. It's no magic at all. Actually, it's very simple and quite clever. ATI does not use a proprietary DVI connector on their cards, contrary to what most people say on internet forums and articles. There are no dedicated audio pins on a standard DVI or HDMI connector, nor does ATI employee any on their connector. DVI and HDMI connectors contain 3 sets of data signals that are used to transmit video data. The manner in which the video data is sent is basically the same for DVI and HDMI (thus why it is so easy to convert one to the other). For HDMI, audio is interleaved between the video data, on the same pins. This is how ATI transmits audio over their DVI port and is also the reason why you don't loose the audio when you use their dongle and convert is back to DVI to HDMI to DVI to HDMI, etc.
So, what does the dongle do? It simply tells the ATI card that it is plugged in and to start sending the audio (again, not over dedicated audio pins but over the 3 sets of data signals). Basically, when the dongle is plugged in, ATI pretends the DVI connector is an HDMI one and electrically treats it like that.